Mind the gap and the map: Measuring distributional and spatial income polarization

Getis-Ord Lecture in Spatial Analysis

Sergio J. Rey

Center for Open Geographical Science

Department of Geography

San Diego State University

February 13, 2025

Overview

  • Polarization
  • Framework
  • Empirical Illustration
  • Conclusion

Polarization

Context

Regional Science Literature

Revenge Narrative

Rodríguez-Pose (2018)

Partisan Sorting

Kaplan, Spenkuch, and Sullivan (2022)

Revenge Realized

Rey and Casimiro Vieyra (2023)

Spatial Polarization

Consensus

  • Important to address

Uncertainty (Pike et al. 2024)

  • How to measure?
  • What index?
  • What scale?
  • How to treat dynamics?

Dynamics and Polarization: Separated at Birth?

Quah (1996)

Wolfson (1994)

\[\begin{equation} P_W = (2G_B - G) \frac{\mu}{m}. \end{equation}\] where \(G_B\) is the Gini index of inequality between the two halves, \(G\) is the Gini index for the entire distribution, \(\mu\) is the mean of the distribution, and \(m\) is the median. Letting \(L(p)\) be the ordinate of the Lorenz curve at the \(p_{th}\) percentile, the discrepancy between perfect equality and the Lorenz curve at the median is: \(D_{50} = 0.5 - L(0.5)\). Given that \(G_B = D_{50}\), the Gini index can be rewritten as \(G=G_B + G_W\) with \(G_W\) representing inequality within the two halves of the distribution. This implies the polarization index is: \[\begin{equation} P_W = (G_B-G_W)\frac{\mu}{m}. \end{equation}\]

Bipolarization

Bipolarization

Components

  • Alienation: Increasing gap between the two halves
  • Group Identification: decrease in local variance

Bipolarization vs. Inequality

  • Not synonyms
  • Pigou-Dalton: progressive income transfer must reduce inequality
  • Same transfer may increase polarization (e.g., Musk to Trump)

Multipolarization (Esteban and Ray 1994)

  • Division into \(k\) ordered groups
  • \(\pi_i= N_i /\sum_i N_i\)
  • \(\mu_i\) is the relative mean income of group \(i\)

\[ P_{ER} (\alpha) = \sum_{i=1}^n \sum_{j=1}^n \pi_{i}^{\alpha+1} \pi_j \left| \mu_i - \mu_j \right| \]

The parameter \(\alpha\) is the degree of sensitivity to polarization. When \(\alpha=0\), \(P_{ER}(\alpha) = 2 G_{B}\), where \(G_B\) is the Gini index between groups.

Spatial Polarization (Zhang and Kanbur 2001)

\[ I(y) = \sum_{g}^K w_g I_g + I(\mu_1 e_1, \mu_2 e_2, \ldots, \mu_k e_k) \] \[ w_g = \begin{cases} f_g \left( \frac{\mu_g}{\mu} \right)^c & \text{if } c \neq 0, 1 \\ f_g \left( \frac{\mu_g}{\mu} \right) & \text{if } c = 1 \\ f_g & \text{if } c = 0. \end{cases} \] \(I_g\) is the inequality in group \(g\), \(\mu_g\) is the mean of incomes for members of group \(g\) and \(e_g\) is a unity vector with \(n_g\) elements, and \(f_g=\frac{n_g}{n}\). The parameter \(c\) measures transfer sensitivity. For \(c < 2\), the measure is more sensitive to transfers in the bottom of the distribution than to those in the top end of the distribution. When \(c=1\) or \(c=0\) the Theil (1972) measure applies.

Spatial Polarization

The measure of spatial polarization is the ratio of the between-group inequality over the within-group inequality: \[ P_{ZK} = \frac{ I(\mu_1 e_1, \mu_2 e_2, \ldots, \mu_k e_k)}{\sum_{g}^K w_g I_g }. \]

The polarization can thus grow with increasing distances between the average incomes of the regions as well as a decrease in the inequality within the regions.

Inspiration

Bipolarization

  • Attribute Bipolarization (Wolfson 1994)
    • Attribute distribution used to define groups
  • Spatial Polarization (Esteban and Ray 1994; Zhang and Kanbur 2001)
    • Regionalization used to define groups
    • Interpret alienation polarization as interregional inequality
    • Interpret identification polarization as intraregional inequality

Integration

  • Combine attribute and spatial polarization
  • Embed in a dynamic framework à la Quah (1996)

Family of Spatial Polarization Measures

Spatial Scale

  • identification of polarization clusters
  • heirarchical scales of polarization (global-meso-local)

Dynamics

  • relax fixed regionalization assumption
  • extend traditional interregional inequality decomposition framework

Framework

Polarization as Graph Intersection

Areas as nodes

  • Each area will be a vertex \(v_i\), \(i=1,2,\ldots, n\)
  • \(|V| = n\)

Two Graphs

  • Distributional (Attribute) Polarization Graph
  • Geographical (Spatial) Graph

Attribute Polarization Graph

\[A = (V_A, E_A)\]

\[E_A = \{(i, j) \in V \times V \mid g(f(i)) = g(f(j))\}\] where \(g(f(i))\) is a function mapping \(i\) to a value class. The specific form of this mapping obtains from the type of polarization under consideration.

Geographical Adjacency Graph

\[G = (V_G, E_G)\] codifies the geographical relationships between each pair of locations.

Components

Both graphs share identical vertex sets, \(V_A = V_G = V\), with each vertex linked to a locational observation \(i\), and \(|V| = n\).

  • Let \(C_G = \{C_{G,1}, C_{G,2}, ..., C_{G,k_{G}}\}\) define the \(k_G\) connected components of \(G\).
  • Similarly, define \(C_A = \{C_{A,1}, C_{A,2}, ..., C_{A,k_{A}}\}\) as the component set for \(A\).
  • Let \(k=max(k_G, k_A)\).

Intersection Graph

\[I=(V_I, E_I)\] where

  • \(V_I=V_G=V_A\)
  • \(E_I = E_G \cap E_A\)

Define \(C_I = \{C_{I,1}, C_{I,2}, \ldots, C_{I,k_{I}}\}\) as the component set for \(I\), with \(k_{I} = \left | C_I \right |\).

Spatial Polarization Index

The index of spatial polarization is: \[ S(G, A) = 1 - \frac{k_I - k}{ (n - k)} \]

  • \(k =max(k_G, k_A)\).

Spatial Bipolarization

Given a set of \(n\) locations with a vector of per capita incomes \(Y\), let \(m= \text{median}(Y)\). Define \(A\) as the \(n \times n\) attribute adjacency graph with edge conditions: \[ A_{ij} = \begin{cases} 1 & \text{if } (Y_i \leq m \text{ and } Y_j \leq m) \text{ or } (Y_i > m \text{ and } Y_j >m), \\ 0 & \text{otherwise}. \end{cases} \]

\(A\) will have \(k_A=2\) components of equal size when \(n\) is even, and approximately equal size when \(n\) is odd.

Define \(G\) as the spatial adjacency graph with: \[ G_{ij} = \begin{cases} 1 & \text{if observations } i \text{ and } j \text{ are geographically adjacent}, \\ 0 & \text{otherwise}. \end{cases} \] The number of connected components for the spatial adjacency graph, \(k_G\), depends on the locational configuration. In a connected spatial adjacency graph, there is a path between each pair of locations, and thus there will be no isolated vertices or disconnected subgraphs. In such a graph \(k_G=1\).

Spatial Bipolarization

\[\begin{equation} S^B(A, G) = 1 - \frac{k_I-k}{n-2}. \label{eq:sp} \end{equation}\] with \(k_I = |C_I|\) , \(I = A \cap G\), and \(k = max(k_A, k_G)\).

\[ 0 \le S^B \le 1\]

\[ S^B = \begin{cases} 1 &k_I = 2 \\ 0 &k_I = n. \end{cases} \]

Spatial Bipolarization

Views

  • Spatial distribution of attribute polarization
  • Polarization of the spatial join structure

Scales of Spatial Bipolarization

\[ V = C_{I,1} \cup C_{I,2} \cup \dots \cup C_{I,k_I} \] with the property that:

\[C_{I,f} \cap C_{I,h} = \emptyset \ \forall \ f \ne h \]

Spatial Polarization Clusters

\[C_{I,f}\]

  • a path exists between each pair of member nodes in the spatial adjacency subgraph induced by the intersection
  • a path exists between each pair of member nodes in the attribute graph

Scales of Spatial Bipolarization

Global

\[S^B\]

Meso

\[C_I\]

Local

\[ C_{I,l} = \{ V_j \in V : V_j \text{ is in component } l \} \]

Meso Scale: \(C_{I,f}\)

For each component \(f\) of graph \(I\), consider the measure:

\[\begin{equation} \zeta_f = \frac{\sum_{i \in f} \sum_j G_{i,j} A_{i,j}}{\sum_{i \in f} \sum_j G_{i,j}}. \end{equation}\]

For the nodes in this intersection component, this represents the proportion of incident edges from the geographical adjacency graph that also exist in the intersection component. In other words, this is the share of each location’s geographical neighbors that belong to the same attribute component as the location.

Local Scale: \(V_i\)

\[ \zeta_i =\frac{\sum_j G_{i,j}A_{i,j}} {\sum_j G_{i,j}} \]

Global-Meso-Local

Global

\[ \zeta = \frac{\sum_i \sum_j G_{i,j} A_{i,j}}{\sum_i \sum_j G_{i,j}} \]

Global from Meso

\[ \zeta = \sum_f \frac{\sum_{i \in f} \sum_j G_{i,j}} {\sum_i \sum_j G_{i,j}} \zeta_f \]

Global from Local

\[\zeta = \sum_i \frac{\sum_j G_{i,j}}{\sum_i \sum_j G_{i,j}}\zeta_i\]

Dual Hierarchy for Attribute Polarization

\[ \nu_f = \frac{\sum_{i \in f} \sum_j G_{i,j} A_{i,j}}{\sum_{i \in f} \sum_j A_{i,j}} \]

and

\[ \nu_i = \frac{\sum_j G_{i,j} A_{i,j}} {\sum_j A_{i,j}} \]

Spatial Multipolarization

Quartile Classification

In a quartile-based grouping there would be four components in the attribute graph and, assuming the spatial graph has a single component, our index of spatial multipolarization becomes: \[ S^M(A, G) = 1 - \frac{k_I-4}{n-4}. \]

Map Complexity (MacEachren 1982)

High and Low Map Complexity

Empirical Illustration

Mexico

States

Spatial Dynamics

Relative Quartiles

Queen Adjacency Graph

Bipolar Adjacency Graph 1940

Spatial Bipolarization Graph

Spatial Bipolarization

\[\begin{equation} S^B(A, G) = 1 - \frac{k_I-k}{n-2} = 1-\frac{6-2}{32-2} = 0.87. \label{e:S2} \end{equation}\]

with \(k_I = |C_I|\) , \(I = A \cap G\), and \(k = max(k_A, k_G)\).

Dynamics of Inequality and Spatial Polarization

Scales of Spatial Polarization

\[\begin{equation} 1 - \frac{k_{i,low} - 1}{n_{low}-1} = 1- \frac{1_{} - 1}{16-1} = 1 \end{equation}\] and for the high end of the distribution we have: \[\begin{equation} 1- \frac{k_{i,high} - 1}{n_{high}-1} = 1 - \frac{5 - 1}{16-1} = 0.73 \end{equation}\] Note that the overall spatial polarization index is the average of these two components.

Local neighborhood homogeneity index Mexico state income 1940.

Local class connectivity index Mexico state income 1940.

Spatial multipolarization graph 1940 (quartiles: Q1 dark blue, Q2 blue, Q3 red, Q4 dark red).

Regimes

Hanson (1996)

Spatial bipolarization intersection graph using the regionalization scheme from Hanson (1996).

Polarization as Regional Inequality Decomposition

Conclusion

Contributions

  • Family of spatial polarization indices
  • Graph based frameing
  • Distinguish between concentrated advantage and concentrated disadvantage
  • Cluster identification
  • Dynamic spatial partitions

Next Steps

  • Comparative spatial polarization (US-EU, US-Mexico, etc.)
  • Properties of polarization index
  • Multivariate extensions
  • pysal-inequality

Thanks

References

Esteban, J. M, and D. Ray. 1994. “On the Measurement of Polarization.” Econometrica 62 (4): 819–51.
Hanson, Gordon H. 1996. U.S.-Mexico Integration and Regional Economies. Cambridge: National Bureau of Economic Research.
Kaplan, Ethan, Jörg L. Spenkuch, and Rebecca Sullivan. 2022. “Partisan Spatial Sorting in the United States: A Theoretical and Empirical Overview.” Journal of Public Economics 211 (July): 104668. https://doi.org/10.1016/j.jpubeco.2022.104668.
MacEachren, Alan M. 1982. “Map Complexity: Comparison and Measurement.” The American Cartographer 9 (1): 31–46. https://doi.org/10.1559/152304082783948286.
Pike, Andy, Vincent Béal, Nicolas Cauchi-Duval, Rachel Franklin, Nadir Kinossian, Thilo Lang, Tim Leibert, et al. 2024. Left Behind Places’: A Geographical Etymology.” Regional Studies 58 (6): 1167–79. https://doi.org/10.1080/00343404.2023.2167972.
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Zhang, Xiaobo, and Ravi Kanbur. 2001. “What Difference Do Polarisation Measures Make?: An Application to China.” Journal of Development Studies 37: 85–98.